Euclid: Photometric redshift calibration with self-organising maps

1. Problem Statement

The Euclid mission aims to map the large-scale structure of the universe up to redshift $z \sim 3$ using weak gravitational lensing (cosmic shear). The accuracy of cosmological constraints (e.g., on dark energy and matter density) depends critically on the precision of the mean redshift ($\langle z \rangle$) for galaxies in tomographic bins.

• The Challenge: Photometric redshifts (photo-zs) suffer from systematic biases and scatter. Calibration relies on spectroscopic samples (spec-z), which are often incomplete or suffer from selection effects that do not match the broader photometric survey.
• The Requirement: Euclid requires the bias in the mean redshift per bin to satisfy $|\Delta\langle z \rangle| \le 0.002(1+z)$.
• The Gap: Previous methods using Self-Organising Maps (SOMs) have shown promise, but it was unclear if they could meet Euclid's stringent requirements under realistic conditions, particularly regarding how redshift tomography (binning) is defined (by cell mean spec-z vs. individual photo-zs) and the impact of calibration sample realism.

2. Methodology

The authors utilized the Euclid Flagship2 simulation (a massive N-body simulation with 2.6 billion galaxies) to create a controlled testing environment.

Data Preparation

• Validation Sample: A large, EWS-like (Euclid Wide Survey) photometric sample ($\sim 1.7$ million galaxies) with noisy photometry (mimicking DR3 depth) and photo-zs estimated via the NNPZ (Nearest Neighbours Photometric Redshifts) algorithm.
• Calibration Sample: A smaller, "gold" spectroscopic sample ($\sim 33,000$ galaxies) generated by selecting galaxies from a noiseless version of the simulation based on SOM cell occupancy. This mimics the non-trivial selection functions of real spectroscopic surveys (like C3R2).
• Sub-sampling: The data was divided into 100 independent line-of-sight sub-fields to assess sample variance.

SOM Architecture & Training

• Two-Stage SOM Approach:
  1. Masters-like SOM (Noiseless): Trained on noiseless photometry (36 colors + $I_E$ magnitude) on a $75 \times 150$ grid. This defines the "feature space" and is used to generate the calibration sample.
  2. Noisy SOM: Trained on the noisy photometry of the generated calibration sample on a $101 \times 101$ toroidal grid. This SOM is then populated with the noisy EWS-like validation sample.
• Quality Control (QC): A "Gold Sample" selection removes cells lacking representation in both samples. An additional QC cut ($|B_p| < 0.5$) removes cells with high photo-z bias to further reduce systematic errors.

Calibration Strategies Compared

The study tested two distinct methods for defining redshift tomography (assigning galaxies to bins):

1. Spec-z Tomography: Assigning galaxies to bins based on the mean spectroscopic redshift of the SOM cell they occupy.
2. Photo-z Tomography: Assigning galaxies to bins based on the individual photo-z point estimate (median of the PDF) of each galaxy, regardless of the cell's mean spec-z.

3. Key Contributions

• Validation of SOMs for Euclid: Demonstrated that SOMs can effectively map high-dimensional color-magnitude space to redshift distributions for the Euclid mission.
• Tomography Definition: Identified that defining redshift bins based on individual galaxy photo-zs is superior to using the mean spec-z of SOM cells. The latter introduces significant bias due to the heterogeneity of galaxy populations within a single SOM cell.
• Realism Scaling: Progressively increased the realism of the calibration sample by morphing it to mimic the C3R2 survey (a real-world spectroscopic survey), testing the robustness of the method under realistic selection functions.
• Cosmological Impact: Performed a Fisher forecast to quantify how residual redshift biases propagate into biases on cosmological parameters ($\Omega_m, \sigma_8, w_0, w_a$).

4. Results

Bias Performance ($\Delta\langle z \rangle$)

• Spec-z Tomography (Cell Mean): Failed to meet Euclid requirements. Even with QC, none of the 10 bins satisfied the $|\Delta\langle z \rangle| \le 0.002(1+z)$ requirement. Biases were often an order of magnitude too large and showed a sign flip (positive at low $z$, negative at high $z$).
• Photo-z Tomography (Individual): Significantly superior.
  • In the idealized scenario: 8 out of 10 bins met the requirement.
  • In the realistic scenario (mimicking C3R2 selection): 6 out of 10 bins met the requirement.
• Conclusion: Defining tomography on individual photo-zs avoids the "smearing" effect caused by averaging diverse populations within a single SOM cell.

Cosmological Parameter Biases

Using a Fisher matrix forecast for cosmic shear only:

• Idealized Case: Biases on cosmological parameters were generally $< 0.1\sigma$.
• Realistic Case: Biases increased but remained $< 0.3\sigma$ for most parameters.
• Parameter Sensitivity: The dark energy equation of state parameter at $z=0$ ($w_0$) was the most sensitive to redshift biases, showing a long tail in its bias distribution, though the mean bias remained below $1\sigma$. The matter density ($\Omega_m$) and amplitude ($\sigma_8$) were less affected.

5. Significance

• Proof of Concept: The paper provides a robust proof of concept that Self-Organising Maps, when combined with appropriate quality control and individual photo-z binning, can meet the stringent calibration requirements of Stage IV cosmological surveys like Euclid.
• Methodological Shift: It challenges the common practice of binning by SOM cell properties, advocating instead for binning by individual galaxy estimates to preserve the fidelity of the redshift distribution.
• Future Roadmap: The study highlights the critical need for extensive, deep spectroscopic calibration samples (like C3R2) to cover under-sampled regions of color space. It sets a baseline for the Euclid Consortium's data analysis pipeline for the DR3 release, suggesting that while current methods are promising, continued refinement of calibration samples is essential to minimize systematic errors in dark energy constraints.

In summary, the paper demonstrates that the Euclid mission can achieve its redshift calibration goals using SOMs, provided the analysis prioritizes individual galaxy photo-zs over cell-averaged spectroscopic values and employs rigorous quality control to mitigate selection biases.